Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo
Abstract
Dedekind's problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number M(n). Backtests at M(8) and M(9) calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision n=9 estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; n=11 and n=13 center-window estimates show a larger-contrast analogous pattern. The protocol estimate for M(10) is M̂(10)=(8.9360±0.0010)× 1078, where the displayed uncertainty is the budget-based forecast scale from the cross-n scaling law under the production budget.
Versions
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BibTeX
@misc{chen2026finiten,
title = {{Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo}},
author = {Tian-Shun Chen and Hao Feng and Haozhe Wang and Chian-Shu Chen and Kilar Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2606.09795},
eprint = {2606.09795},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.09795},
}
RIS
TY - UNPB
TI - Finite-n Estimate of Dedekind Numbers by Layer-Ratio Monte Carlo
AU - Tian-Shun Chen
AU - Hao Feng
AU - Haozhe Wang
AU - Chian-Shu Chen
AU - Kilar Zhang
PY - 2026
DA - 2026/08/03
PB - arXiv
DO - 10.48550/arxiv.2606.09795
UR - https://arxiv.org/abs/2606.09795
AB - Dedekind's problem counts monotone Boolean functions, equivalently downsets of a Boolean lattice. We recast this enumeration as a finite layer-ratio reconstruction problem for the Whitney numbers of the ranked ideal lattice. An exact adjacent-layer double count expresses each layer ratio through local averages of the number of addable elements and the number of removable elements. Reversible fixed-layer Markov chains estimate these averages and hence estimate the Dedekind number M(n). Backtests at M(8) and M(9) calibrate seed-level variability under the fixed protocol and measure the observed Monte Carlo budget scaling. The resulting estimate probes the Whitney-number sequence of the ideal lattice. Although these rows have previously been described empirically as unimodal, the high-precision n=9 estimate has a shallow two-shoulder feature around the central rank, contrary to that empirical description; n=11 and n=13 center-window estimates show a larger-contrast analogous pattern. The protocol estimate for M(10) is M̂(10)=(8.9360±0.0010)× 10^78, where the displayed uncertainty is the budget-based forecast scale from the cross-n scaling law under the production budget.
ER -